Final announcement

This is to inform the noble audience of the course that the main program of the course is completed. I will stay in Pisa for one more week (till January 8, 2015) and will be happy to discuss any subject (upon request).

Meanwhile one of the subjects discussed in this course was brought to a pre-final form: the manuscript

1. Shira Tanny, Sergei Yakovenko, On local Weyl equivalence of higher order Fucshian equations, arXiv:1412.7830,

was posted on arXiv and submitted to the Arnold Mathematical Journal, a new venue for publications molded in the spirit of  the late V. I. Arnol’d and his seminar.

Any criticism will be most appreciated. Congratulations modestly accepted.

Tanti auguri, carissimi! Buon anno, happy New Year, с наступающим Новым Годом, שנה (אזרחית) טובח, bonne année!

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Lecture 12+ (Mon, Dec 22, 2014)

Riemann-Hilbert problem

The Riemann-Hilbert problem consists in “constructing a Fuchsian system with a prescribed monodromy”.

More precisely, let be nondegenerate matrices such that their product is an identical matrix, and are distinct points, such that the segments are all disjoint except for the point itself.

The problem is to construct a linear system of equations

,

such that the monodromy operator along the path “segment small loop around segment ” is equal to .

The modern strategy of solving this problem is surgery. One can easily construct a local solution, a differential system on a neighborhood of the segment , which has the specified monodromy. The phase space of this system is the cylinder , and without loss of generality one can assume that together the neighborhoods cover the whole Riemann sphere . Patching together these local solutions, one can construct a linear system with the specified monodromy, but it will be defined not on , as required, but on a more general object, holomorphic vector bundle over .

Description of different vector bundles is of an independent interest and is well known. It turns out (Birkhoff), that each holomorphic vector bundle in dimension is completely determined by a(n unordered) tuple of integer numbers , and the bundle is trivial if and only if .

However, the strategy of solving the Riemann-Hilbert problem by construction of the bundle and determining its holomorphic type is complicated by two facts:

1. Determination of the holomorphic type of a bundle is a transcendental problem;
2. The local realization of the monodromy is by no means unique: in the non-resonant case one can realize any matrix by an Euler system with the eigenvalues which can be arbitrarily shifted by integers; in the resonant case one should add to this freedom also non-Euler systems. This freedom can change the holomorphic type of the vector bundle in a very broad range.

It turns out that the fundamental role in solvability of the Riemann-Hilbert problem plays the (ir)reducibility of the linear group generated by the matrices .

Theorem (Bolibruch, Kostov). If the group is irredicible, i.e., there is no invariant subspace in common for all operators , then one can choose the local realizations in such a way that the resulting bundle is trivial and thus yields solution to the Riemann-Hilbert problem. 

The proof is achieved as follows: one constructs a possibly nontrivial bundle realizing the given monodromy, and then this bundle is brutally trivialized by a transformation that is only meromorphic at one of the singularities. The result will be a system with all but one singularities being Fuchsian, and the problem reduces to bringing to the Fuchsian form the last point (assumed to be at infinity) by transformations of the form with being a matrix polynomial with a constant nonzero determinant.  The group of such transformations is considerably more subtle, but ultimately the freedom in construction of the initial bundle can be used to guarantee that the last point is also “Fuchsianizible”.

All the way around, if the monodromy group is reducible, then there is an obstruction of the torsion type exists for trivializing the bundle. This obstruction was first discovered by A. Bolibruch, and its description can be found in the textbook by Yu. Ilyashenko and SY (sections 16G and 18).

Lecture 12 (Friday Dec. 19, 2014)

Noetherian chains

Computation of the (local intersectional) degree of a phase curve of a polynomial vector field, produced in Lecture 11, is based on the length of the ascending chain of polynomial ideals generated by consecutive derivations.

Let be the Lie derivation of the algebra of polynomials along the vector field . It increases the degrees by at most . Let be a seed polynomial of degree and consider the ascending chain of ideals

where .  By Noetherianity, this chain must eventually stabilize at some step: . In addition to this chain of ideals, one can consider the associated descending chain of algebraic varieties

This chain also stabilizes  no later than on the th step, but may stabilize earlier.  The following properties of these chains can be verified by elementary arguments.

1. The chain of ideals is strictly ascending: If , then all subsequent ideals in the chain coincide.
2. The chain of varieties may be nonstrictly ascending: e.g., .
3. The length of the descending chain measures the maximal nontrivial order of contact between the trajectories of and the hypersurface .

In general, the length of a strictly ascending chain of polynomial ideals generated by the sequence of polynomials of degrees not exceeding an explicit (growing) function of , can be bounded by an algorithmically computable function. However, even in the simplest case where (as above), this function turns out to be the Ackermann generalized exponential, a recursive but not primitively recursive function of which grows faster than any elementary (or primitive recursive) function. It is the algebraic origin of the sequence of polynomials, which allows to establish better results.

Example. Assume that is an endomorphism of the ring of the polynomials, and instead of the iterations of the Lie derivation, we consider the sequence . Then analogous chains can be constructed, yet their properties will be slightly different (in a sense, better). In particular, the chain of varieties becomes strictly descending and its length can be relatively simply bounded by simple function of . If the growth rate of is linear (as above), the bound will be double exponential in . However,  in general the growth rate of iterates is exponential, which leads to the bound given by a tower function (iterated exponent) of height .

The easiest way to estimate the length of varieties generated by consecutive derivations is based on the explicit Nullstellensatz. By this  theorem, for any polynomial which vanishes on the variety which is the zero locus of an ideal there exist a finite power such that . The number can be explicitly bounded from above (J. Kollar, 1988): if is generated by polynomials of degree no greater than , then . Having this bound, for each irreducible component of the variety which does not belong to the stable limit, one can predict, how many steps in can survive before being eliminated.  The resulting upper bound will be double exponential in .

However, a better, more realistic and simple exponential in upper bound can be achieved by completely different argument.

Example. Assume that and we look at an isolated contact between a (nonsingular) trajectory of a vector field and an algebraic curve of degree at a point . Consider the local analytic chart in which is parallel to the -axis and the point is at the origin. If the curve has tangency of order with the vertical axis, then its projection on the -axis is locally a ramified covering of order . Consider a small bidisk neighborhood of the origin and apply a small analytic perturbation to . The multiple tangency point will be scattered into several points of simple (quadratic) tangency, while the topological covering property will persist. Denote by the number of obtained simple tangencies: at each tangency exactly two leaves of the covering “collide”. Thus the total number of leaves cannot be greater than . The problem thus becomes to estimate . However, the set of points of quadratic contact is algebraic: it is defined by the equations of degrees and , so by the Bezout theorem the number of points does not exceed the product of these two numbers.

To generalize this argument for the multidimensional settings, one has to modify the topological part of the argument dealing with “exactly two leaves of the covering collide”. Instead of just one set , one has to consider the sets (recall that is the dimension of the ambient space), and instead of counting points, one should consider their Euler characteristics. The corresponding combinatorics can be elegantly expressed by the “integration over the Euler characteristic” discovered by O. Viro (1988), while the bounds for the Euler characteristic of algebraic varieties can be bounded by virtue of the J. Milnor’s result (1964).

The result, due to A. Gabrielov and A. Khovanskii, is simple exponential (in ) bound, was achieved in 1998. However, for some problems in the analytic number theory (algebraic independence of transcendental numbers) it is important to have a more precise estimate of the maximal tangency order for fixed, but variable and growing to infinity. The most recent achievements in this direction are due to G. Binyamini [4], see below.

Besides, a different (and considerably more difficult) problem arises in the singular context, when one tries to estimate the order of contact of an algebraic hypersurface with a separatrix of a polynomial vector field, an invariant analytic curve (usually non-smooth) which contains a singular point of the vector field .  Here again the most recent breakthroughs are due to Binyamini [5].

References (in addition to those mentioned earlier).

1. A. Gabrielov, A. Khovanksii, Multiplicity of a Noetherian intersection.  Geometry of differential equations, 119–130,
   Amer. Math. Soc. Transl. Ser. 2, 186, Amer. Math. Soc., Providence, RI, 1998.
2. O. Viro, Some integral calculus based on Euler characteristic. Topology and geometry—Rohlin Seminar, 127–138,
   Lecture Notes in Math., 1346, Springer, Berlin, 1988.
3. J. Milnor, On the Betti numbers of real varieties.  Proc. Amer. Math. Soc. 15 1964 275–280.
4. G. Binyamini, Multiplicity Estimates: a Morse-theoretic approach, arXiv:1406.1858 (2014).
5. G. Binyamini, Multiplicity estimates, analytic cycles and Newton polytopes, arXiv:1407.1183

Lecture 11 (Mon, Dec 15, 2014)

Trajectories of polynomial vector fields

Definition
Let be a finite piece of a real analytic curve. Its intersection complexity (or intersection degree) is the maximal number of isolated intersections of with real affine hyperplanes,

The goal of this lecture is to explain the following result (D. Novikov, S.Y., 1999) which claims that a sufficiently small piece of a nonsingular trajectory of a polynomial vector field has a finite intersection degree bounded in terms of the dimension and the degree of the field.

More specifically, we consider the polynomial vector field associated with the system of polynomial differential equations

.

Denote its integral trajectory passing through an arbitrary point by .

Theorem
For any a sufficiently small piece of has the intersection degree not exceeding .

References.

1. D. Novikov and S. Yakovenko, Trajectories of polynomial vector fields and ascending chains of polynomial ideals, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 2, 563–609, MR1697373 (2001h:32054)
2. S. Yakovenko, Quantitative theory of ordinary dierential equations and the tangential Hilbert 16th problem, On finiteness in differential equations and Diophantine geometry, CRM Monogr. Ser., vol. 24, Amer. Math. Soc., Providence, RI, 2005, pp. 41–109. MR2180125 (2006g:34062)

Lecture 10 (Fri, Dec 12, 2014)

Uniform bounds for parametric Fuchsian families

The previous lectures indicate how zeros of solutions can be counted for linear differential equations on the Riemann sphere. For an equation of the form

one has to assume that:

1. The equation has only Fuchsian singularities at the poles of the coefficients ;
2. The monodromy of each singular point is quasiunipotent (i.e., all eigenvalues of the corresponding operator are on the unit circle);
3. The slope of the differential equation is known.

The slope is a badly formed and poorly computable number that characterizes the relative strength of the non-principal coefficients of the equation. It is defined as follows:

1. For a given affine chart on , multiply the equation (*) by the common denominator of the fractions for , reducing the corresponding operator to the form with ;
2. Define the affine slope as the , where the norm of a polynomial is the sum ;
3. Define the conformal slope of an equation (*) as the supremum of the affine slopes of the corresponding operators over all affine charts on .
4. Claim. If the equation (*) is Fuchsian, then the conformal slope is finite.

The rationale behind the notion of the conformal slope of an equation is simple: it is assumed to be the sole parameter which allows to place an upper bound for the variation of arguments along “simple arcs” (say, circular arcs and line segments) which are away from the singular locus of the equation (*).

The dual notion is the conformal diameter of the singular locus. This is another badly computable but still controllable way to subdivide points of the singular locus into confluent groups that stay away from each other. The formal definition involves the sum of relative lengths of circular slits.

The claim (that is proved by similar arguments as the precious claim on boundedness of the conformal slope) is that a finite set points of the Riemann sphere has conformal diameter bounded. Moreover, if is an algebraic divisor of degree in the -dimensional projective space, then the conformal diameter of any finite intersection
for any 1-dimensional line is explicitly bounded in terms of .

Together these results allow to prove the following general result.

Theorem (G. Binyamini, D. Novikov, S.Y.)

Consider a Pfaffian -system on the projective space with the rational matrix 1-form of degree . Assume that:

1. The system is integrable, ;
2. The system is regular, i.e., its solution matrix grows at worst polynomially when tends to the polar locus
   of the system;
3. The monodromy of the system along any small loop around is quasiunipotent.

Then the number of solutions of any solution is bounded in any triangle free from points of $late \varSigma$.

If in addition the system is defined over and has bitlength complexity $c$, then this number is explicitly bounded by a double exponential of the form , where is an explicit polynomial of degree in these variables.

Remark. The quasiunipotence condition can be verified only for small loops around the principal (smooth) strata of by the Kashiwara theorem.

Reference

G. Binyamini, D. Novikov, and S. Yakovenko, On the number of zeros of Abelian integrals: A constructive solution of the infinitesimal Hilbert sixteenth problem, Inventiones Mathematicae 181 (2010), no. 2, 227-289, available here.

Lecture 9 (Mon., Dec. 1, 2014)

Families of Fuchsian equations

A Fuchsian equation on with only quasiunipotent singularities admits an upper bound for the number of complex roots of its solutions, which depends on the equation, in particular, in the “magnitude” (slope), but also on the relative position of its singularities.

We are interested in finding conditions ensuring that this bound does not “explode”. The easiest way to formulate this is to consider parametric families of Fuchsian equations.

We will assume that the parametric family has the form

with the coefficients polynomial in and rationally depending on the parameters (one can consider them as homogeneous polynomials of the same degree on ). For some values of the operator may degenerate (the leading coefficient vanishes identically, not excluding the degeneracy ). Such values, however, should constitute a proper algebraic subvariety .

Note that, because of the semicontinuity, it is sufficient to establish the global uniform bound for the number of isolated roots only for : complex roots cannot disappear in the blue sky…

We will impose the following qualitative conditions, imposed on the family (*).

1. Isomonodromy: when parameters change, the monodromy group remains “the same”.
2. Tameness (regularity): solutions of the equations grow at most polynomially when .
3. Quasiunipotence: all singular points always have quasiunipotent monodromy.

The last condition is the “regularity” with respect to the parameters rather than with respect to the independent variable . All conditions need to be accurately formulated, but one can give a simple example producing such families.

Consider a rational matrix-valued 1-form on with the polar locus which is an algebraic divisor (singular hypersurface). Assume that the linear system is locally solvable and regular on . Then for any fixed the first row components of the (multivalued) matrix function satisfy a linear Fuchsian equation rationally depending on . This way we get the family of equations automatically satisfying the first two conditions above. It turns out that the third condition is sufficient to verify only for a generic equation of the family.

(Kashiwara theorem follows).

Boundedness of the slope

In the arbitrary family (*) the slope is a semialgebraic function of the parameter , eventually undefined on the locus itself, and may well be unbounded.

However, in the isomonodromic regular family this is impossible.

(Grigoriev theorem follows)

Corollary: conformal pseudoinvariance of the slope.

Lecture 8 (Nov. 26, 2014)

Oscillatory behavior of Fuchsian equations

Semilocal theory

Consider a holomorphic linear equation in the unit disk , having a unique Fuchsian singularity at the origin . Such an equation can be always reduced to the form , with holomorphic bounded coefficients , , .

The previous results imply that one can produce an explicit upper bound for the variation of argument of any nontrivial solution of the equation along the boundary of the unit disk : for some universal constant .

If the solution itself is holomorphic (e.g., in the case of apparent singularities), such bound would imply (by virtue of the argument principle) a bound for the number of zeros of in . Unfortunately, solutions are usually ramified and the argument principle does not work. Denote by the monodromy operator along the boundary.

Definition

The Fuchsian point is called quasiunipotent, if all eigenvalues of the matrix have modulus one, .

Theorem 1

The number of isolated roots of any solution of the equation in the Riemann domain having real coefficients and a single quasiunipotent singularity at the origin does not exceed , where is the parameter bounding the magnitude of coefficients of .

The proof is based on a version of the flavor of the Rolle theorem for the “difference operators” for any unit such that :

A version of the Cayley-Hamilton theorem asserts that the (commutative) composition over all eigenvalues of the monodromy operator (counted with their multiplicities) vanishes on all solutions of the real Fuchsian equation.

Global theory

A linear ordinary differential equatuib with rational coefficients from can always be transformed to the form

It may depend on additional parameters : if this dependence is rational, then we may assume that the coefficients of the operator are polynomials from . The new feature then will be appearance of singular perturbations: for some values of the parameters the leading coefficient may vanish identically in , meaning that the order of the corresponding equation drops down to a smaller value. Such phenomenon is known to cause numerous troubles of analytic nature.

Changing the independent variable allows to investigate the nature of singularity at the infinite point . The equation is called Fuchsian, if it is Fuchsian at each its singular point on the Riemann sphere .

Assume that infinity is non-singular (this can always be achieved by a Mobius transformation of the independent variable ). Then a Fuchsian equation with the singular locus can always be transformed to the form , where is the operator

(nonsingularity at infinity implies certain bounds on the degrees of the polynomials ). However, the coefficients of this form depend in the rational way not only on the coefficients of the original equation (*), but also on the location of the points .

Definition.

The slope of this operator (*) is defined as the maximum

 where the norm of a polynomial is the sum .

Simple inequalities:

1. Any polynomial of known degree and norm admits an explicit upper bound for on any disk : for .
2. A polynomial of unit norm admits a lower bound for for points distant from its zero locus . More precisely,

We expect that for an equation having only quasiunipotent Fuchsian singular points, the number of isolated roots of solutions can be explicitly bounded in terms of and . Indeed, it looks like we can cut out circular neighborhoods of all singularities and apply Theorem 1.

The trouble occurs when singularities are allowed to collide or almost collide. Then any slit separating them will necessarily pass through the area where the leading coefficient is dangerously small.

Schedule change

The next lecture is moved from its usual Friday time slot to Wednesday (tomorrow), November 26, 10:00. This is one-time move.

Lecture 7 (Nov. 24)

Geometric form of non-oscillation theorems

Solutions of linear systems can be highly oscillating relatively to hyperplanes . However, there exists a class of system for which one can produce such bounds.

Let be a smooth parametrized curve. Its osculating frame is the tuple of vector functions (velocity), (acceleration), till . Generically these vectors are linear independent for all except isolated points. The differential equations defining the curve up to a rigid motion have a “companion form”,

Note that this is a vector ODE with scalar coefficients, i.e., a tuple of identical scalar ODEs. Besides, it may exhibit singularities: if the osculating frame degenerates (which may well happen at isolated points of the curve), the coefficients of this equation exhibit a pole at the corresponding moments of time .

However, the osculating frame is not a natural object: it depends on the parametrization. The invariant notion is the osculating flag, the flag of subspaces spanned (in ) by the vectors . The flag can be naturally parametrized by the orthogonalization procedure applied to the osculating frame: by construction, this means that we consider the -tuple of orthonormal vectors with the property that

This new frame satisfies the Frenet equations: their structure follows from the invariance of the flag and the orthogonality of the frame.

The functions are called Frenet curvatures: they are nonnegative except for the last one (hypertorsion) which has sign and may change it at isolated hyperinflection points.

---

Definitions. (Absolute) integral curvatures of a smooth (say, real analytic) curve , parametrized by the arclength , are the quantities , , and (the last quality, equal to the number of hyperinflection points up to the constant , is called integral hyperinflection).

Let be a smooth curve avoiding the origin in the space. Its absolute rotation around the origin is defined as the length of its spherical projection on the unit sphere, .  The absolute rotation around any other point is defined by translating this point to the origin.

If is a -dimensional affine subspace disjoint from and the orthogonal projection on the orthogonal complement , the absolute rotation of $\latex \Gamma$ around is the absolute rotation of the curve around the point .

The absolute rotation of around an affine hyperplane is defined as .

Formally the 0-sphere is not connected, but it is convenient to make it into the metric space with two “antipodal” points at the distance , similarly to higher dimensional unit spheres with antipodal points always distanced at .

Denote by the supremum , where the supremum is taken over all affine subspaces of dimension $k$ in .

---

Main Theorem.

.

The proof of this theorem is based on a combination of arguments from integral geometry and the Frobenius formula for a differential operator vanishing on given, say, real analytic functions . Denote by the Wronski determinant of the first functions , adding for convenience . These Wronskians are real analytic, and assuming that does not vanish identically, we can construct the linear th order differential operator

One can instantly see that this operator is monic (composition of monic operators of order 1) and by induction prove that it vanishes on all functions .

The straightforward application of the Rolle theorem guarantees that if all the Wronskians are nonvanishing on , then the operator is disconjugate and no linear combination of functions can have more than isolated root.

In the case where the Wronskians are allowed to have isolated roots, numbering if counted with multiplicity, then the maximal number of zeros that a linear combination as above may exhibit, is bounded by .

 

References.

1. A. Khovanskii, S. Yakovenko, Generalized Rolle theorem in and . Contains detailed description of the so called Voorhoeve index, the total variation of argument of an analytic function on the boundary of its domain and why this serves as a substitute for the Rolle theorem over the complex numbers. As a corollary, rather sharp bounds for the number of complex roots of quasipolynomials , in complex domains are obtained.
2. D. Novikov, S. Yakovenko, Integral curvatures, oscillation and rotation of smooth curves around affine subspaces. Contains the proof of the Main theorem cited below, with a slightly worse weights attached to the integral curvatures.
3. D. Nadler, S. Yakovenko, Oscillation and boundary curvature of holomorphic curves in . A complex analytic version of the Main theorem with improved estimates.

Lecture 6 (Nov. 21, 2014)

Zeros of solutions of linear equations

Nontrivial (i.e., not identically zero) solutions of linear ordinary differential equations obviously possess certain properties concerning their roots (points where these solutions vanish). The simplest, in a sense paradigmal property, is the following.

Prototheorem. Let be a nontrivial solution of a sufficiently regular linear ordinary differential equation of order . Then cannot have a root of multiplicity greater or equal than .

Here by regularity we mean the condition that the operator has coefficients smooth enough to guarantee that any solution near any point in the domain of its definition is uniquely determined by the initial conditions .

Indeed, if has a root of multiplicity , that is, all first derivatives of at vanish, then by virtue of the equation and hence the by the uniqueness must be identically zero.

In particular, solutions of first order equation are nonvanishing, solutions of any second order equation may have only simple roots etc.

---

What about quantitative versions?

Theorem (de la Vallee Poussin, 1929). Assume that the coefficients of the LODE

are explicitly bounded,  .

Assume that the bounds are small relative to the length of the interval, i.e.,

Then any nontrivial solution of the equation has no more than isolated roots on .

Novikov’s counterexample

What about linear systems of the first order?

Consider the system with and the norm explicitly bounded on . Consider all possible linear combinations . Can one expect a uniform upper bound for the number of roots of all combinations?

Let be a polynomial having many zeros on . Consider the -system of the form

The first equation defines a nonvanishing function , the second equation – its derivative which vanishes at all roots of .

By replacing by one can achieve an arbitrarily small sup-norm of the coefficients of this system on the segment (or even any open complex neighborhood of this real segment). Thus no matter how small are the coefficients, the second component will have the specified number of isolated roots.

---

Complexification

What about complex valued versions? There is no Rolle theorem for them.

I will describe three possible replacements, Kim’s theorem (1963), nearest in the spirit, and two versions of the argument principle.

Theorem (W. Kim)
Assume that an analytic LODE

is defined in a convex compact subset of diameter and the condition (*) holds. Then this equation is disconjugate in : any solution has at most isolated roots.

This result follows from the interpolation inequality of the following type: if is a function holomorphic in and has isolated roots there, then (the maximum modulus norm is assumed).

---

Consider the equation on the real interval but with complex-valued coefficients (and solutions). Solutions will be then real parameterized curves which only exceptionally rarely have roots. Instead of counting roots, one can measure their rotation around the origin , which is defined as for any continuous choice of the argument.

Theorem. Assume that

Then rotation of any nontrivial solution is explicitly bounded: .

If an analytic LODE with explicitly bounded coefficients is defined, say, on a triangle , then application of this result to the sides of the triangle yields an explicit upper bound for the number of isolated roots of analytic solutions inside the triangle.

Reference

S. Yakovenko, On functions and curves defined by differential equations, §2.